Is there an RPG where I can play as the type of elf who lives in a hollow tree and makes cookies?
Seriously though, the first time I read The Lord of the Rings was years before Peter Jackson's adaptation was even announced. So I was still in the mindset that didn't make a distinction between "elves" and "fairies", so in my mind the elves were all as small as hobbits, or even smaller.
The thing is, Tolkien elves are basically Norse elves, who are kind of like air elementals in human-like form, whereas most Celtic-rooted definitions of "elf" and "fairy" are a lot broader, ranging from very Norse (humanoid, beautiful in a vaguely spooky way) to more twee and Victorian (brownies and whatnot) to straight-up freaky (demon centaurs, flaming black bulls), because the term is really just a catchall for everything that isn't a god that lives on the Other Side.
It's a bit clearer cut in ancient Irish tradition than it is in, say, Japanese tradition, although in both there is, in essence, a certain delineation between the sort of divine entities which do things like control storm systems and make the sun rise and lesser powers which inhabit places like mountains and shrines. There is a further difference between the gods and the sidhe in Celtic lore, however, in that gods tend to be more human-like and approachable, representing as they do the aspects of nature which we accept and for the most part understand, whereas the sidhe are more alien and feral, being of the unknown.
The word for "god" in Japanese is pretty vague and generic and applies to both greater and lesser deities, so technically the latter case applies to all of the above.
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although by the Celtic metric all the Centralian fairies are more god than fairy, since they're each associated with a particular, defined attribute